Joint overnight leader with Kevin Streelman, the gritty 42-year-old bogeyed his first and last holes, but was rock-solid in between as he rolled in five birdies to finish with a eight-under total of 134, one ahead of the American.
"I kind of patiently worked my way through the round," European Tour veteran Bjorn told reporters.
"On the back nine, I played awesome... Maybe a bit unfortunate (on) the last hole."
Already uncomfortably fast in Thursday's opening round, Royal Melbourne's greens were rendered devilish during a second straight day of warm sunshine and persistent breezes.
Seemingly well-directed approach shots charged past the holes and players sweated over three-foot putts with a trepidation bordering on fear.
Along with Bjorn, Streelman was among the few to handle the conditions well and charged out of the blocks to birdie his opening four holes.
He had the wind taken out of his sails, however, when he dropped three strokes in two holes midway through his round.
Following his first bogey on the eighth, Streelman pulled his second shot into a greenside bunker on the long par-four ninth and ended up with a double-bogey after missing a putt from 12 feet.
"I knew as soon as it went right of that pin I was toast," Streelman said of his eight-iron approach.
"That's the mistake you can't make around this golf course and I know better than that and it bit me."
Australia's Jason Day (70) and Portugal's Ricardo Santos (69) were tied for third, four strokes behind Bjorn, with American Matt Kuchar (68) among a group of four players a further stroke adrift.
Sixty players are competing for individual honours for the first time at the World Cup, which was previously solely a team tournament.
Twenty-six two-man teams of compatriots are also competing, with the best aggregate scores after four rounds of strokeplay determining the winning nation.
Kuchar's solid three-under round helped propel the United States into the lead for team honours on 10-under, three strokes ahead of Denmark's Bjorn and Thorbjorn Olesen.
Australia's Day and Adam Scott were tied for third with Japan's Hideto Tanihara (67) and Ryo Ishikawa (71), a further four strokes adrift.
A day after being punished with a quintuple-bogey nine at the 12th hole, world number two Scott carded a solid three-under 68 but remains nine strokes behind Bjorn.
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- Kevin Streelman